Domanda del libro di testo
Without using a calculator, find the exact value of: [log3 81 - log𝝅 1]/[log2√2 8 - log 0.001]
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Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 5, Problema 144
Without using a calculator, find the exact value of log4 [log3 (log₂ 8)].
Guida verificata passo dopo passo1
Start by evaluating the innermost logarithm: \( \log_2 8 \). Recall that \( \log_b a = c \) means \( b^c = a \). Since \( 2^3 = 8 \), we have \( \log_2 8 = 3 \).
Next, substitute this value into the next logarithm: \( \log_3 (\log_2 8) = \log_3 3 \). Using the same definition, since \( 3^1 = 3 \), it follows that \( \log_3 3 = 1 \).
Now, substitute this result into the outermost logarithm: \( \log_4 [\log_3 (\log_2 8)] = \log_4 1 \).
Recall that for any base \( b > 0 \) and \( b \neq 1 \), \( \log_b 1 = 0 \) because \( b^0 = 1 \). Therefore, \( \log_4 1 = 0 \).
Thus, the exact value of the original expression \( \log_4 [\log_3 (\log_2 8)] \) is \( 0 \).
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Change of Base and Nested Logarithms
Understanding how to evaluate nested logarithms requires recognizing the order of operations and simplifying from the innermost logarithm outward. Each logarithm must be evaluated exactly before applying the next, ensuring clarity in the base and argument at each step.
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5:36
Change of Base Property
Evaluating Logarithms with Simple Arguments
Logarithms with arguments that are powers of the base can be simplified using the identity log_b(b^k) = k. For example, log₂ 8 simplifies to 3 because 8 = 2^3. This simplification is key to finding exact values without a calculator.
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5:14Evaluate Logarithms
Properties of Logarithms and Exact Values
Knowing logarithm properties, such as log_b(1) = 0 and log_b(b) = 1, helps in simplifying expressions. Exact values are found by expressing numbers as powers of the base and applying these properties step-by-step to avoid approximations.
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5:36Change of Base Property
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